Memoirs on Differential Equations and Mathematical Physics
Volume 51, 2010, 93–108
Said Kouachi and Belgacem Rebiai
INVARIANT REGIONS AND
THE GLOBAL EXISTENCE
FOR REACTION-DIFFUSION SYSTEMS
WITH A TRIDIAGONAL MATRIX
OF DIFFUSION COEFFICIENTS
Abstract. The aim of this study is to prove the global existence of
solutions for reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions. In so doing, we
make use of the appropriate techniques which are based on invariant domains and Lyapunov functional methods. The nonlinear reaction term has
been supposed to be of polynomial growth. This result is a continuation of
that by Kouachi [12].
2010 Mathematics Subject Classification. 35K45, 35K57.
Key words and phrases. reaction diffusion systems, invariant domains, Lyapunov functionals, global existence.
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Reaction-Diffusion Systems
95
1. Introduction
We consider the reaction-diffusion system
∂u
− a11 ∆u − a12 ∆v − a23 ∆w = f (u, v, w) in R+ × Ω,
∂t
∂v
− a21 ∆u − a22 ∆v − a23 ∆w = g(u, v, w) in R+ × Ω,
∂t
∂w
− a21 ∆u − a32 ∆v − a11 ∆w = h(u, v, w) in R+ × Ω,
∂t
(1.1)
(1.2)
(1.3)
with the boundary conditions
∂u
= β1 ,
∂η
∂v
λv + (1 − λ)
= β2 on R+ × ∂Ω,
∂η
∂w
λw + (1 − λ)
= β3 ,
∂η
λu + (1 − λ)
(1.4)
and the initial data
u(0, x) = u0 (x),
v(0, x) = v0 (x),
w(0, x) = w0 (x) in Ω,
(1.5)
where:
(i) 0 < λ < 1 and βi ∈ R, i = 1, 2, 3, for nonhomogeneous Robin
boundary conditions.
(ii) λ = βi = 0, i = 1, 2, 3, for homogeneous Neumann boundary conditions.
(iii) 1 − λ = βi = 0, i = 1, 2, 3, for homogeneous Dirichlet boundary
conditions.
Ω is an open bounded domain of the class C1 in RN with boundary ∂Ω, and
∂
∂η denotes the outward normal derivative on ∂Ω. The diffusion terms aij
(i, j = 1, 2, 3 and (i, j) 6= (1, 3), (3, 1)) are supposed to be positive constants
with a11 = a33 and (a12 + a21 )2 + (a23 + a32 )2 < 4a11 a22 , which reflects the
parabolicity of the system and implies at the same time that the matrix of
diffusion


a11 a12 0
A = a21 a22 a23 
0 a32 a11
is positive definite. The eigenvalues λ1 , λ2 and λ3 (λ1 < λ2 , λ3 = a11 ) of A
are positive. If we put
a = min {a11 , a22 } and a = max {a11 , a22 } ,
then the positivity of aij ’s implies that
λ1 < a ≤ λ3 ≤ a < λ 2 .
96
Said Kouachi and Belgacem Rebiai
The initial data are assumed to be in the domain
o
n
3

(u
,
v
,
w
)
∈
R
:
µ
v
≤
a
u
+
a
w
≤
µ
v
,
a
u
≤
a
w
0
0
0
2
0
21
0
23
0
1
0
32
0
12
0





n if µ2 β2 ≤ a21 β1 + a23 β3 ≤ µ1 β2 , a32 β1 ≤ a12 β3 ,
o



3

(u
,
v
,
w
)
∈
R
:
µ
v
≤
a
u
+
a
w
≤
µ
v
,
a
w
≤
a
u
0
0
0
2
0
21
0
23
0
1
0
12
0
32
0







 if µ2 β2 ≤ a21 β1 + a23 β3 ≤ µ1 β2 , a12 β3 ≤ a32 β1 ,




(u0 , v0 , w0 ) ∈ R3 :




1
1
Σ =  µ2 (a21 u0 +a23 w0 ) ≤ v0 ≤ µ1 (a21 u0 +a23 w0 ) , a32 u0 ≤ a12 w0 


1
1



(a21 β1 +a23 β3 ) ≤ β2 ≤
(a21 β1 +a23 β3 ) , a32 β1 ≤ a12 β3 ,
if


µ2
µ1




3

(u0 , v0 , w0 ) ∈ R :





1
1


(a21 u0 +a23 w0 ) ≤ v0 ≤
(a21 u0 +a23 w0 ) , a32 u0 ≥ a12 w0 



µ2
µ1



1
1


if
(a21 β1 +a23 β3 ) ≤ β2 ≤
(a21 β1 +a23 β3 ) , a32 β1 ≥ a12 β3 ,
µ2
µ1
where
µ1 = a − λ1 > 0 > µ2 = a − λ2 .
Since we use the same methods to treat all the cases, we will tackle only
with the first one. We suppose that the reaction terms f, g and h are
continuously differentiable, polynomially bounded on Σ,
³
´
f (r1 , r2 , r3 ) , g (r1 , r2 , r3 ) , h (r1 , r2 , r3 )
is in Σ for all (r1 , r2 , r3 ) in ∂Σ (we say that (f, g, h) points into Σ on ∂Σ),
i.e.,
a21 f (r1 , r2 , r3 ) + a23 h (r1 , r2 , r3 ) ≤ µ1 g (r1 , r2 , r3 )
(1.6)
for all r1 , r2 and r3 such that µ2 r2 ≤ a21 r1 +a23 r3 = µ1 r2 and a32 r1 ≤ a12 r3 ,
and
µ2 g (r1 , r2 , r3 ) ≤ a21 f (r1 , r2 , r3 ) + a23 h (r1 , r2 , r3 )
(1.6a)
for all r1 , r2 and r3 such that µ2 r2 = a21 r1 +a23 r3 ≤ µ1 r2 and a32 r1 ≤ a12 r3 ,
and
a32 f (r1 , r2 , r3 ) ≤ a12 h (r1 , r2 , r3 )
(1.6b)
for all r1 , r2 and r3 such that µ2 r2 ≤ a21 r1 +a23 r3 ≤ µ1 r2 and a32 r1 = a12 r3 ,
and for positive constants E and D, we have
(Ef + Dg + h) (u, v, w) ≤ C1 (u + v + w + 1),
(1.7)
for all (u, v, w) in Σ, where C1 is a positive constant.
In the two-component case, where a12 = 0, Kouachi and Youkana [13]
generalized the method of Haraux and Youkana [4] with the reaction terms
f (u, v) = −λF (u, v) and g (u, v) = µF (u, v) with F (u, v) ≥ 0, requiring
the condition
¸
·
ln (1 + F (r, s))
< α∗ for any r ≥ 0,
lim
s→+∞
s
Reaction-Diffusion Systems
97
with
∗
α =
2a11 a22
2
n (a11 − a22 ) ku0 k∞
½
min
λ a11 − a22
,
µ
a21
¾
,
where the positive diffusion coefficients a11 , a22 satisfy a11 > a22 , and a21 ,
λ, µ are positive constants. This condition reflects the weak exponential
growth of the reaction term F . Kanel and Kirane [6] proved the global
existence in the case where g (u, v) = −f (u, v) = uv n and n is an odd
integer, under the embarrassing condition
|a12 − a21 | < Cp ,
where Cp contains a constant from Solonnikov’s estimate [18]. Later they
improved their results in [7] to obtain the global existence under the restrictions
H1 . a22 < a11 + a21 ,
a11 a22 (a11 + a21 − a22 )
if a11 ≤ a22 < a11 + a21 ,
H2 . a12 < ε0 =
a11 a22 + a21 (a11 + a21 − a22 )
½
¾
1
H3 . a12 < min
(a11 + a21 ) , ε0 if a22 < a11 ,
2
and
³
´
1−ε
|F (v)| ≤ CF 1 + |v|
, vF (v) ≥ 0 for all v ∈ R,
where ε and CF are positive constants with ε < 1 and
g (u, v) = −f (u, v) = uF (v) .
Kouachi [12] has proved global existence for solutions of two-component
reaction-diffusion systems with a general full matrix of diffusion coefficients
and nonhomgeneous boundary conditions.
Many chemical and biological operations are described by reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients. The components u (t, x), v (t, x) and w (t, x) can represent either chemical concentrations or biological population densities (see, e.g., Cussler [1] and [2]).
We note that the case of strongly coupled systems which are not triangular in the diffusion part is more difficult. As a consequence of the blow-up
of the solutions found in [16], we can indeed prove that there is a blow-up of
the solutions in finite time for such nontriangular systems even though the
initial data are regular, the solutions are positive and the nonlinear terms
are negative, a structure that ensured the global existence in the diagonal
case. For this purpose, we construct invariant domains in which we can
demonstrate that for any initial data in these domains, the problem (1.1)–
(1.5) is equivalent to the problem for which the global existence follows
from the usual techniques based on Lyapunov functionals (see Kirane and
Kouachi [8], Kouachi and Youkana [13] and Kouachi [12]).
98
Said Kouachi and Belgacem Rebiai
2. Local Existence and Invariant Regions
This section is devoted to proving that if (f, g, h) points into Σ on ∂Σ,
then Σ is an invariant domain for the problem (1.1)–(1.5), i.e., the solution
remains in Σ for any initial data in Σ. Once the invariant domains are
constructed, both problems of the local and global existence become easier
to be established. For the first problem we demonstrate that the system
(1.1)–(1.3) with the boundary conditions (1.4) and the initial data in Σ is
equivalent to a problem for which the local existence throughout the time
interval [0, T ∗ [ can be obtained by the known procedure, and for the second
one we need invariant domains as explained in the preceeding section.
The main result of this section is
Proposition 1. Suppose that (f, g, h) points into Σ on ∂Σ. Then for any
(u0 , v0 , w0 ) in Σ the solution (u, v, w) of the problem (1.1)–(1.5) remains in
Σ for all t’s in [0, T ∗ [ .
t
Proof. Let (xi1 , xi2 , xi3 ) , i = 1, 2, 3, be the eigenvectors of the matrix At
associated with its eigenvalues λi , i = 1, 2, 3 (λ1 < λ3 < λ2 ). Multiplying
the equations (1.1), (1.2) and (1.3) of the given reaction-diffusion system by
xi1 , xi2 and xi3 , respectively, and summing the resulting equations, we get
∂
z1 − λ1 ∆z1 = F1 (z1 , z2 , z3 ) in ]0, T ∗ [ × Ω,
∂t
∂
z2 − λ2 ∆z2 = F2 (z1 , z2 , z3 ) in ]0, T ∗ [ × Ω,
∂t
∂
z3 − λ3 ∆z3 = F3 (z1 , z2 , z3 ) in ]0, T ∗ [ × Ω,
∂t
(2.1)
(2.2)
(2.3)
with the boundary conditions
λzi + (1 − λ)
∂zi
= ρi , i = 1, 2, 3, on ]0, T ∗ [ × ∂Ω,
∂η
(2.4)
and the initial data
zi (0, x) = zi0 (x), i = 1, 2, 3, in Ω,
(2.5)
zi = xi1 u + xi2 v + xi3 w, i = 1, 2, 3, in ]0, T ∗ [ × Ω,
(2.6)
where
ρi = xi1 β1 + xi2 β2 + xi3 β3 , i = 1, 2, 3,
and
Fi (z1 , z2 , z3 ) = xi1 f + xi2 g + xi3 h, i = 1, 2, 3,
(2.7)
for all (u, v, w) in Σ.
First, as has been mentioned above, note that the condition of the parabolicity of the system (1.1)–(1.3) implies the parabolicity of the system
Reaction-Diffusion Systems
99
(2.1)–(2.3) since
2
2
(a12 + a21 ) + (a23 + a32 ) < 4a11 a22 =⇒
=⇒ (det A > 0 and a11 a22 − a23 a32 > 0).
Since λ1 , λ2 and λ3 (λ1 < λ3 < λ2 ) are the eigenvalues of the matrix At , the
problem (1.1)–(1.5) is equivalent to the problem (2.1)–(2.5) and to prove
that Σ is an invariant domain for the system (1.1)–(1.3) it suffices to prove
that the domain
ª ¡ ¢3
© ¡ 0 0 0¢
(2.8)
z1 , z2 , z3 ∈ R3 : zi0 ≥ 0, i = 1, 2, 3 = R+
is invariant for the system (2.1)–(2.3) and that
o
n
Σ = (u0 , v0 , w0 ) ∈ R3 : zi0 = xi1 u0 +xi2 v0 +xi3 w0 ≥ 0, i = 1, 2, 3 .
(2.9)
t
Since (xi1 , xi2 , xi3 ) is an eigenvector of the matrix At associated to the
eigenvalue λi , i = 1, 2, 3, we have
(a11 − λi ) xi1 + a21 xi2 = 0,
a12 xi1 + (a22 − λi ) xi2 + a32 xi3 = 0 , i = 1, 2, 3,
a23 xi2 + (a11 − λi ) xi3 = 0.
If we assume, without loss of generality, that a11 ≤ a22 and choose x12 = µ1 ,
x22 = −µ2 and x33 = a12 , then we have


−a21 u0 +µ1 v0 −a23 w0 ≥ 0,
⇐⇒
xi1 u0+xi2 v0+xi3 w0 ≥ 0, i = 1, 2, 3 ⇐⇒ a21 u0 −µ2 v0 +a23 w0 ≥ 0,


−a32 u0 + a12 w0 ≥ 0.
⇐⇒ µ2 v0 ≤ a21 u0 + a23 w0 ≤ µ1 v0 ,
Thus (2.9) is proved and (2.6) can be written as


z1 = −a21 u + µ1 v − a23 w,
z2 = a21 u − µ2 v + a23 w,


z3 = −a32 u + a12 w.
a32 u0 ≤ a12 w0 .
(2.6a)
3
Now, to prove that the domain (R+ ) is invariant for the system (2.1)–(2.3),
it suffices to show that Fi (z1 , z2 , z3 ) ≥ 0 for all (z1 , z2 , z3 ) such that zi = 0
and zj ≥ 0, j = 1, 2, 3 (j 6= i) , i = 1, 2, 3, thanks to the invariant domain
method (see Smoller [17]). Using the expressions (2.7), we get


F1 = −a21 f + µ1 g − a23 h,
(2.7a)
F2 = a21 f − µ2 g + a23 h,


F3 = −a32 f + a12 h
for all (u, v, w) in Σ. Since from (1.6), (1.6a) and (1.6b) we have Fi (z1 , z2 , z3 )
≥ 0 for all (z1 , z2 , z3 ) such that zi = 0 and zj ≥ 0, j = 1, 2, 3 (j 6= i) ,
i = 1, 2, 3, we obtain zi (t, x) ≥ 0, i = 1, 2, 3, for all (t, x) ∈ [0, T ∗ [ × Ω.
Then Σ is an invariant domain for the system (1.1)–(1.3).
¤
100
Said Kouachi and Belgacem Rebiai
In addition, the system (1.1)–(1.3) with the boundary conditions (1.4)
and initial data in Σ is equivalent to the system (2.1)–(2.3) with the boundary conditions (2.4) and positive initial data (2.5). As has been mentioned
at the beginning of this section and since ρi , i = 1, 2, 3, given by


ρ1 = −a21 β1 + µ1 β2 − a23 β3 ,
ρ2 = a21 β1 − µ2 β2 + a23 β3 ,


ρ3 = −a32 β1 + a12 β3 ,
¡ ¢
are positive, we have for any initial data in C Ω or Lp (Ω), p ∈ ]1, +∞] ,
the local existence and uniqueness of solutions to the initial value problem
(2.1)–(2.5) and consequently those of the problem (1.1)–(1.5) follow from
the basic existence theory for abstract semilinear differential equations (see
Friedman [3], Henry [5] and Pazy [15]). These solutions are classical on
[0, T ∗ [ × Ω, where T ∗ denotes the eventual blow up time in L∞ (Ω). A local
solution is continued globally by a priori estimates.
Once invariant domains are constructed, one can apply the Lyapunov
technique and establish the global existence of unique solutions for (1.1)–
(1.5).
3. Global Existence
As the determinant of the linear algebraic system (2.6), with respect to
the variables u, v and w, is different from zero, to prove the global existence
of solutions of the problem (1.1)–(1.5) one needs to prove it for the problem
(2.1)–(2.5). To this end, it suffices (see Henry [5]) to derive a uniform
estimate of kFi (z1 , z2 , z3 )kp , i = 1, 2, 3 on [0, T ], T < T ∗ , for some p > N/2,
where k · kp denotes the usual norms in spaces Lp (Ω) defined by
Z
1
p
p
kukp =
|u(x)| dx, 1 ≤ p < ∞, and kuk∞ = esssup |u(x)| .
|Ω|
x∈Ω
Ω
Let θ and σ be two positive constants such that
θ > A12 ,
¡ 2
¢¡ 2
¢
2
2
2
θ − A12 σ − A23 > (A13 − A12 A23 ) ,
(3.1)
(3.2)
where
λi + λj
Aij = p
, i, j = 1, 2, 3 (i < j) ,
2 λi λj
and let
2
2
θq = θ(p−q+1) and σp = σ p , for q = 0, 1, . . . , p and p = 0, 1, . . . , n, (3.3)
where n is a positive integer. The main result of this section is
Reaction-Diffusion Systems
101
Theorem 1. Let (z1 (t, ·) , z2 (t, ·) , z3 (t, ·)) be any positive solution of
(2.1)–(2.5). Introduce the functional
Z
¡
¢
(3.4)
t 7−→ L(t) = Hn z1 (t, x) , z2 (t, x) , z3 (t, x) dx,
Ω
where
Hn (z1 , z2 , z3 ) =
p
n X
X
Cnp Cpq θq σp z1q z2p−q z3n−p ,
(3.5)
p=0 q=0
n!
.
with n being a positive integer and Cnp = (n−p)!p!
Then the functional L is uniformly bounded on the interval [0, T ], T < T ∗ .
For the proof of Theorem 1 we need some preparatory Lemmas.
Lemma 1. Let Hn be the homogeneous polynomial defined by (3.5).
Then
p
n−1
XX
∂Hn
(n−1)−p
p
=n
Cn−1
Cpq θq+1 σp+1 z1q z2p−q z3
,
∂z1
p=0 q=0
(3.6)
p
n−1
XX
∂Hn
(n−1)−p
p
,
Cn−1
Cpq θq σp+1 z1q z2p−q z3
=n
∂z2
p=0 q=0
(3.7)
p
n−1
XX
∂Hn
(n−1)−p
p
Cn−1
Cpq θq σp z1q z2p−q z3
.
=n
∂z3
p=0 q=0
(3.8)
Proof. Differentiating Hn with respect to z1 and using the fact that
q−1
p−1
qCpq = pCp−1
and pCnp = nCn−1
(3.9)
for q = 1, 2, . . . , p, p = 1, 2, . . . , n, we get
p
n X
X
∂Hn
p−1 q−1
Cn−1
Cp−1 θq σp z1q−1 z2p−q z3n−p .
=n
∂z1
p=1 q=1
Replacing in the sums the indexes q − 1 by q and p − 1 by p, we deduce
(3.6). For the formula (3.7), differentiating Hn with respect to z2 , taking
into account
Cpq = Cpp−q , q = 0, 1, . . . , p − 1 and p = 1, 2, . . . , n,
(3.10)
using (3.9) and replacing the index p − 1 by p, we get (3.7).
Finally, we have
p
n−1
XX
∂Hn
=
(n − p) Cnp Cpq θq σp z1q z2p−q z3n−p−1 .
∂z3
p=0 q=0
n−p−1
p
Since (n − p) Cnp = (n − p) Cnn−p = nCn−1
= nCn−1
, we get (3.8).
¤
102
Said Kouachi and Belgacem Rebiai
Lemma 2. The second partial derivatives of Hn are given by
p
n−2
XX
∂ 2 Hn
(n−2)−p
p
=
n
(n
−
1)
Cn−2
Cpq θq+2 σp+2 z1q z2p−q z3
,
∂z12
p=0 q=0
(3.11)
p
n−2
XX
∂ 2 Hn
(n−2)−p
p
Cpq θq+1 σp+2 z1q z2p−q z3
,
= n (n − 1)
Cn−2
∂z1 ∂z2
p=0 q=0
(3.12)
p
n−2
XX
∂ 2 Hn
(n−2)−p
p
= n (n − 1)
Cn−2
Cpq θq+1 σp+1 z1q z2p−q z3
,
∂z1 ∂z3
p=0 q=0
(3.13)
p
n−2
XX
∂ 2 Hn
(n−2)−p
p
=
n
(n
−
1)
Cn−2
Cpq θq σp+2 z1q z2p−q z3
,
∂z22
p=0 q=0
(3.14)
p
n−2
XX
∂ 2 Hn
(n−2)−p
p
= n (n − 1)
Cn−2
Cpq θq σp+1 z1q z2p−q z3
,
∂z2 ∂z3
p=0 q=0
(3.15)
p
n−2
XX
∂ 2 Hn
(n−2)−p
p
=
n
(n
−
1)
Cn−2
Cpq θq σp z1q z2p−q z3
.
∂z32
p=0 q=0
Proof. Differentiating
(3.16)
∂Hn
given by (3.6) with respect to z1 yields
∂z1
p
n−1
XX
∂ 2 Hn
(n−1)−p
p
.
qCn−1
Cpq θq+1 σq+1 z1q−1 z2p−q z3
=
n
∂z12
p=1 q=1
Using (3.9), we get (3.11).
p−1
n−1
XX
∂ 2 Hn
(n−1)−p
p
.
Cpq θq+1 σp+1 z1q z2p−q−1 z3
(p − q) Cn−1
=n
∂z1 ∂z2
p=1 q=0
Applying (3.10) and then (3.9), we get (3.12).
p
n−2
XX
∂ 2 Hn
(n−2)−p
p
((n − 1) − p) Cn−1
Cpq θq+1 σp+1 z1q z2p−q z3
.
=n
∂z1 ∂z3
p=0 q=0
Applying successively (3.10), (3.9) and (3.10) for the second time, we deduce
(3.13).
p−1
n−1
XX
∂ 2 Hn
(n−1)−p
p
=
n
(p − q) Cn−1
Cpq θq σp+1 z1q z2p−q−1 z3
.
∂z22
p=1 q=0
The application of (3.10) and then of (3.9) yields (3.14).
p
n−2
XX
∂ 2 Hn
(n−2)−p
p
=n
((n − 1) − p) Cn−1
Cpq θq σp z1q z2p−q z3
.
∂z2 ∂z3
p=0 q=0
Reaction-Diffusion Systems
103
Applying (3.10) and then (3.9) yields (3.15). Finally we get (3.16) by dif∂Hn
ferentiating
with respect to z3 and applying successively (3.10), (3.9)
∂z3
and (3.10) for the second time.
¤
Proof of Theorem 1. Differentiating L with respect to t yields
¶
Z µ
∂Hn ∂z1
∂Hn ∂z2
∂Hn ∂z3
0
L (t) =
+
+
dx =
∂z1 ∂t
∂z2 ∂t
∂z3 ∂t
Ω
¶
Z µ
∂Hn
∂Hn
∂Hn
∆z1 + λ2
∆z2 + λ3
∆z3 dx+
=
λ1
∂z1
∂z2
∂z3
Ω
¶
Z µ
∂Hn
∂Hn
∂Hn
+
F1 +
F2 +
F3 dx =
∂z1
∂z2
∂z3
Ω
=: I + J.
Using Green’s formula in I, we get I = I1 + I2 , where
¶
Z µ
∂Hn ∂z1
∂Hn ∂z2
∂Hn ∂z3
I1 =
λ1
+ λ2
+ λ3
ds,
∂z1 ∂η
∂z2 ∂η
∂z3 ∂η
∂Ω
where ds denotes the (n − 1)-dimensional surface element, and
Z ·
∂ 2 Hn
∂ 2 Hn
2
|∇z
|
+
(λ
+
λ
)
∇z1 ∇z2
I2 = −
λ1
1
1
2
∂z12
∂z1 ∂z2
Ω
∂ 2 Hn
∇z1 ∇z3 + λ2
∂z1 ∂z3
∂ 2 Hn
+ (λ2 + λ3 )
∇z2 ∇z3 + λ3
∂z2 ∂z3
+ (λ1 + λ3 )
∂ 2 Hn
2
|∇z2 |
∂z22
¸
∂ 2 Hn
2
|∇z3 | dx.
∂z32
We prove that there exists a positive constant C2 independent of t ∈ [0, T ∗ [
such that
I1 ≤ C2 for all t ∈ [0, T ∗ [ ,
(3.17)
and that
I2 ≤ 0.
(3.18)
To see this, we follow the same reasoning as in [11].
(i) If 0 < λ < 1, using the boundary conditions (2.4) we get
¶
Z µ
∂Hn
∂Hn
∂Hn
I1 =
λ1
(γ1 −αz1 )+λ2
(γ2 −αz2 )+λ3
(γ3 −αz3 ) ds,
∂z1
∂z2
∂z3
∂Ω
where α =
λ
1−λ
and γi =
ρi
1−λ ,
i = 1, 2, 3. Since
∂Hn
∂Hn
∂Hn
(γ1 −αz1 )+λ2
(γ2 −αz2 )+λ3
(γ3 −αz3 )
∂z1
∂z2
∂z3
= Pn−1 (z1 , z2 , z3 ) − Qn (z1 , z2 , z3 ) ,
H (z1 , z2 , z3 ) = λ1
104
Said Kouachi and Belgacem Rebiai
where Pn−1 and Qn are polynomials with positive coefficients and respective
degrees n − 1 and n, and since the solution is positive, we obtain
lim sup
H (z1 , z2 , z3 ) = −∞,
(3.19)
(|z1 |+|z2 |+|z3 |)→+∞
3
which proves that H is uniformly bounded on (R+ ) , and consequently
(3.17).
(ii) If λ = 0, then I1 = 0 on [0, T ∗ [ .
(iii) The case of the homogeneous Dirichlet conditions is trivial since the
∂z2
∂z3
1
positivity of the solution on [0, T ∗ [×Ω implies ∂z
∂η ≤ 0, ∂η ≤ 0 and ∂η ≤ 0
on [0, T ∗ [ × ∂Ω. Consequently, one again gets (3.17) with C2 = 0.
Now, we prove (3.18). Applying Lemma 1 and Lemma 2, we get
Z n−2
p
XX
£
¤
p
I2 = −n (n − 1)
Cn−2
Cpq (Bpq z) · z dx,
Ω p=0 q=0
where
Bpq

λ1 θq+2 σp+2

 λ1 + λ2
=
θq+1 σp+2

2
λ +
λ3
1
θq+1 σp+1
2
λ1 + λ2
θq+1 σp+2
2
λ2 θq σp+2
λ2 + λ3
θq σp+1
2

λ1 + λ3
θq+1 σp+1

2

λ2 + λ3
θq σp+1 
,
2

λ3 θq σp
t
for q = 0, 1, . . . , p, p = 0, 1, . . . , n − 2 and z = (∇z1 , ∇z2 , ∇z3 ) .
The quadratic forms (with respect to ∇z1 , ∇z2 and ∇z3 ) associated with
the matrices Bpq , q = 0, 1, . . . , p, p = 0, 1, . . . , n − 2, are positive since
their main determinants ∆1 , ∆2 and ∆3 are positive too, according to the
Sylvester criterion. To see this, we have
1. ∆1 = λ1 θq+2 σp+2 > 0 for q = 0, 1, . . . , p and p = 0, 1, . . . , n − 2.
¯
¯
¯
¯
λ1 + λ2
¯ λ1 θq+2 σp+2
θq+1 σp+2 ¯¯
¯
2
2. ∆2 = ¯ λ + λ
¯
2
¯ 1
¯
θq+1 σp+2
λ2 θq σp+2
¯
¯
2
¡
¢
2
2
θ2 − A212 ,
= λ1 λ2 θq+1
σp+2
for q = 0, 1, . . . , p and p = 0, 1, . . . , n − 2.
Using (3.1), we get ∆2 > 0.
¯
¯
¯
¯
λ1 + λ2
λ1 + λ3
¯ λ1 θq+2 σp+2
θq+1 σp+2
θq+1 σp+1 ¯
¯
¯
2
2
¯ λ1 + λ2
¯
λ2 + λ3
¯
3. ∆3 = ¯
θq+1 σp+2
λ2 θq σp+2
θq σp+1 ¯¯
2
2
¯
¯
λ2 + λ3
¯ λ1 + λ3
¯
θq+1 σp+1
θq σp+1
λ3 θq σp
¯
¯
2
2
£
¤
2
2
= λ1 λ2 λ3 θq+1
θq σp+2 σp+1
(θ2−A212 )(σ 2−A223 )−(A13−A12 A23 )2 ,
for q = 0, 1, . . . , p and p = 0, 1, . . . , n − 2.
Reaction-Diffusion Systems
105
Using (3.2), we get ∆3 > 0. Consequently we have (3.18).
Substitution of the expressions of the partial derivatives given by Lemma
1 in the second integral yields
#
Z " n−1
p
XX
p
q q p−q (n−1)−p
J=
n
Cn−1 Cp z1 z2 z3
×
Ω
p=0 q=0
× (θq+1 σp+1 F1 + θq σp+1 F2 + θq σp F3 ) dx.
Using the expressions (2.7a), we get
θq+1 σp+1 F1 + θq σp+1 F2 + θq σp F3 =
¡
¢
= − θq+1 σp+1 a21 +a21 θq σp+1 −a32 θq σp f +(θq+1 σp+1 µ1 − µ2 θq σp+1 ) g+
¡
¢
+ − θq+1 σp+1 a23 + a23 θq σp+1 + a12 θq σp h =
³
´µ a (θ σ
21
q p+1 −θq+1 σp+1 )−a32 θq σp
= a23 (θq σp+1−θq+1 σp+1 )+a12 θq σp
f+
a23 (θq σp+1 −θq+1 σp+1 )+a12 θq σp
¶
θq+1 σp+1 µ1 − µ2 θq σp+1
g+h =
+
a23 (θq σp+1 − θq+1 σp+1 ) + a12 θq σp
µ
µ
¶
¶
σp+1
θq
θq
= θq+1 σp a23
−1 +a12
×
σp
θq+1
θq+1


¡ θq
¢
θq
θq σp+1
σ
σ
− µ2 θq+1
µ1 σp+1
a21 σp+1
θq+1 − 1 −a32 θq+1
σp
p
p
×
f+
g + h .
¡ θq
¢
¢
σ
θq
σp+1 ¡ θq
θq
a23 σp+1
−
1
+a
a
−
1
+
a
12
23
12
θq+1
θq+1
σp
θq+1
θq+1
p
θ
σ
q
Since θq+1
and σp+1
are sufficiently large if we choose θ and σ sufficiently
p
large, using the condition (1.7) and the relation (2.6a) successively we get,
for an appropriate constant C3 ,
#
Z " n−1
p
XX
p
q q p−q (n−1)−p
dx.
(z1 + z2 + z3 + 1) Cn−1 Cp z1 z2 z3
J ≤ C3
p=0 q=0
Ω
To prove that the functional L is uniformly bounded on the interval [0, T ],
we first write
p
n−1
XX
(n−1)−p
p
(z1 + z2 + z3 + 1) Cn−1
Cpq z1q z2p−q z3
=
p=0 q=0
= Rn (z1 , z2 , z3 ) + Sn−1 (z1 , z2 , z3 ) ,
where Rn (z1 , z2 , z3 ) and Sn−1 (z1 , z2 , z3 ) are two homogeneous polynomials
of degrees n and n − 1, respectively. First, since the polynomials Hn and
Rn are of degree n, there exists a positive constant C4 such that
Z
Z
Rn (z1 , z2 , z3 ) dx ≤ C4 Hn (z1 , z2 , z3 ) dx.
Ω
Ω
106
Said Kouachi and Belgacem Rebiai
Applying Hölder’s inequality to the integral
Z
R
Ω
Sn−1 (z1 , z2 , z3 ) dx, one gets

 n−1
n
Z
n
n−1
dx
.
Sn−1 (z1 , z2 , z3 ) dx ≤ (meas Ω)  (Sn−1 (z1 , z2 , z3 ))
1
n
Ω
Ω
Since for all z1 ≥ 0 and z2 , z3 > 0
n
n
(Sn−1 (ξ1 , ξ2 , 1)) n−1
(Sn−1 (z1 , z2 , z3 )) n−1
=
,
Hn (z1 , z2 , z3 )
Hn (ξ1 , ξ2 , 1)
where ξ1 =
z1
z2 ,
ξ2 =
z2
z3
and
n
(Sn−1 (ξ1 , ξ2 , 1)) n−1
< +∞,
ξ1 →+∞
Hn (ξ1 , ξ2 , 1)
lim
ξ2 →+∞
one asserts that there exists a positive constant C5 such that
n
(Sn−1 (z1 , z2 , z3 )) n−1
≤ C5 for all z1 , z2 , z3 ≥ 0.
Hn (z1 , z2 , z3 )
Hence the functional L satisfies the differential inequality
L0 (t) ≤ C6 L (t) + C7 L
n−1
n
(t) ,
1
which for Z = L n can be written as
nZ 0 ≤ C6 Z + C7 .
A simple integration gives a uniform bound of the functional L on the
interval [0, T ]. This completes the proof of Theorem 1.
¤
Corollary 1. Suppose that the functions f (r1 , r2 , r3 ), g (r1 , r2 , r3 ) and
h (r1 , r2 , r3 ) are continuously differentiable on Σ, point into Σ on ∂Σ and
satisfy the condition (1.7). Then all uniformly bounded on Ω solutions of
(1.1)–(1.5) with the initial data in Σ are in L∞ (0, T ; Lp (Ω)) for all p ≥ 1.
Proof. The proof of this Corollary is an immediate consequence of Theorem 1, the trivial inequality
Z
p
(z1 + z2 + z3 ) dx ≤ L (t) on [0, T ∗ [ ,
Ω
and (2.6a).
¤
Proposition 2. Under the hypothesis of Corollary 1, if f (r1 , r2 , r3 ),
g (r1 , r2 , r3 ) and h (r1 , r2 , r3 ) are polynomially bounded, then all uniformly
bounded on Ω solutions of (1.1)–(1.4) with the initial data in Σ are global
in time.
Reaction-Diffusion Systems
107
Proof. As has been mentioned above, it suffices to derive a uniform estimate
of kF1 (z1 , z2 , z3 )kp , kF2 (z1 , z2 , z3 )kp and kF3 (z1 , z2 , z3 )kp on [0, T ], T < T ∗
for some p > N2 . Since the reactions f (u, v, w), g (u, v, w) and h (u, v, w) are
polynomially bounded on Σ, by using relations (2.6a) and (2.7a) we get that
so are F1 (z1 , z2 , z3 ), F2 (z1 , z2 , z3 ) and F3 (z1 , z2 , z3 ), and the proof becomes
an immediate consequence of Corollary 1.
¤
Acknowledgments
The authors would like to thank the anonymous referee for his/her valuable comments and suggestions that helped to improve the presentation of
this paper.
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(Received 22.01.2008; revised 23.09.2009)
Authors’ addresses:
Said Kouachi
Department of Mathematics
College of Science
Qassim University
P.O.Box 6644, Al-Gassim, Buraydah 51452
Saudi Arabia
E-mail: kouachi.said@caramil.com
Belgacem Rebiai
Department of Mathematics
University of Tebessa, 12002
Algeria
E-mail: brebiai@gmail.com